Does not converge nor does it diverge to infinity or negative infinity

Question

I am stuck on part D of my problem. Suppose that $a_n$ converges to $0$ and $b_n$ converges to infinity. $c_n = (a_n) \times (b_n)$, Give an example where $(c_n)$ does not converge, nor does it diverge to infinity or negative infinity

Answer 1

How about $\displaystyle a_n=\frac{1}{n}$. Can you find $b_n$?

Answer 2

$((-1)^n/n)\rightarrow 0$ and $(n)\rightarrow \infty$ what about the product?